Section 2.9: Exponential Functions
An exponential function is a function in which the variable appears as an exponent: \[ f(x) = a \cdot b^x \] where \(a \neq 0\) is the initial value and \(b > 0, b \neq 1\) is the base.
Exponential functions model growth and decay:
- Growth: \( b > 1 \)
- Decay: \( 0 < b < 1 \)
Example 1: Growth
Population grows according to \( P(t) = 500 \cdot 1.05^t \). Find population after 3 years.
\( P(3) = 500 \cdot 1.05^3 = 500 \cdot 1.157625 \approx 578.81 \) Population ≈ 579
Example 2: Decay
Radioactive substance decays: \( N(t) = 200 \cdot (0.9)^t \). Find remaining after 5 hours.
\( N(5) = 200 \cdot 0.9^5 = 200 \cdot 0.59049 \approx 118.10 \) Remaining ≈ 118 units
Practice Problems
- Evaluate \( f(4) \) for \( f(x) = 3 \cdot 2^x \)
- Determine if \( f(x) = 100 \cdot 0.8^x \) represents growth or decay
- Find \( t \) when \( N(t) = 50 \) for \( N(t) = 200 \cdot 0.9^t \)
- Write an exponential function for a bacteria culture that doubles every hour, initial = 10
- Graph \( f(x) = 2 \cdot (0.5)^x \) for x = 0 to 5